Write the equation for a parabola with a focus at $(-4,3)$ and a directrix at $y=5$. $y=$
Solution: The strategy A parabola is defined as the set of all points that are the same distance away from a point (the focus) and a line (the directrix). Let $(x,y)$ be a point on the parabola. Then the distance between $(x,y)$ and the focus, $(-4,3)$, is equal to the distance between $(x,y)$ and the directrix, $y=5$. Once we find these distances, we can equate them in order to derive the equation of our parabola. Finding the distances from $(x,y)$ to the focus and the directrix The distance between $(x,y)$ and $(-4,3)$ is $\sqrt{(x+4)^2+(y-3)^2}$. [How did we find that?] Similarly, the distance between $(x,y)$ and the line $y=5$ is $\sqrt{(y-5)^2}$. [How did we know that?] Deriving the formula by equating the distances $\begin{aligned} \sqrt{(y-5)^2} &= \sqrt{(x+4)^2+(y-3)^2} \\\\ (y-5)^2 &= (x+4)^2+(y-3)^2 \\\\ {y^2}-10y{+25} &= (x+4)^2{+y^2}{-6y}+9\\\\ -10y{+6y}&=(x+4)^2+9{-25} \\\\ -4y&=(x+4)^2-16 \\\\ y&=-\dfrac{(x+4)^2}{4}+4\end{aligned}$ The answer The equation of our parabola is $y=-\dfrac{(x+4)^2}{4}+4$. Here is the graph of our parabola. As expected, the distance between a point on the parabola, $(x,y)$, and the focus is the same as the distance between $(x,y)$ and the directrix. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ $y$ $x$ ${(x,y)}$